Quantum Numbers — n, l, m_l and m_s Explained

Q: What are the four quantum numbers?

n is the principal quantum number (energy level), l is the azimuthal quantum number (orbital shape), m_l is the magnetic quantum number (orbital orientation), and m_s is the spin quantum number (+1/2 or −1/2). Together they label a single electron's complete state in an atom.

Q: What quantum numbers are possible for n = 2?

n = 2 allows l = 0 (the 2s subshell) and l = 1 (2p). For l = 0 there is only m_l = 0. For l = 1, m_l = −1, 0, +1 — three p orbitals. With two spin states each, the n = 2 shell holds 2 + 6 = 8 electrons, which is why the periodic table's second row has eight elements.

Q: Why is the spin quantum number only ±1/2?

Spin is quantized like other angular momenta. Electrons are spin-1/2 fermions, so the component of spin measured along any axis can only be ±ħ/2 — written ±1/2 in units of ħ. Other particles differ: photons are spin-1, for example.

Q: Do protons and neutrons have quantum numbers?

Yes — nucleons in a nucleus occupy shell-model states with their own quantum numbers, and the same Pauli principle applies. It explains the 'magic numbers' of extra-stable nuclei such as helium-4 and lead-208.

Introductory College⏱ 10 min readModern Physics

The four quantum numbers (n, l, m_l, m_s) that describe every electron: allowed values, the Pauli exclusion principle, and how they build the periodic table.

Quick answer

Quantum numbers are four integers or half-integers — n, l, m_l, m_s — that completely describe an electron's state in an atom: its energy level, its orbital's shape, the orbital's orientation in space, and the electron's spin.

Example: Hydrogen's ground-state electron is fully described by n = 1, l = 0, m_l = 0, and m_s = +1/2 or −1/2. With n = 1, no other l or m_l values are even possible.

What you will learn

Name the four quantum numbers and state what physical property each describes
List the allowed values of l, m_l and m_s for a given n
Write a valid set of quantum numbers for any electron in a light atom
Use the Pauli exclusion principle to explain why two electrons can share an orbital

Why electrons need addresses

Solving Schrödinger's equation for hydrogen does not give one answer — it gives an infinite family of solutions, each describing a different allowed state of the electron. Four numbers label every solution, the way a street, city and apartment number pinpoint one address.

These quantum numbers — $n$ , $l$ , $m_l$ and $m_s$ — are not arbitrary bookkeeping. Each corresponds to a real, measurable property: energy, angular momentum, orientation, and spin. Every electron in every atom in the universe carries a set of four.

n: the principal quantum number

The principal quantum number $n = 1, 2, 3, \dots$ sets the electron's energy level and the rough size of its orbital — the 'shell'. Larger $n$ means a larger, higher-energy orbital.

For hydrogen, the energies form a beautifully simple ladder:

E_n = \frac{-13.6\ \mathrm{eV}}{n^2}

n = 1 is the ground state; n → ∞ approaches zero (a free electron)

l: the azimuthal quantum number

The azimuthal quantum number $l$ sets the orbital's shape and the electron's orbital angular momentum. It can take integer values from $0$ up to $n - 1$ — so an $n = 3$ shell has $l = 0, 1, 2$ .

Chemists use letters from old spectroscopy: $l = 0$ is an s orbital (spherical), $l = 1$ is p (dumbbell), $l = 2$ is d (cloverleaf-like), $l = 3$ is f (complex). The letters originally stood for sharp, principal, diffuse and fundamental spectral lines.

m_l: the magnetic quantum number

The magnetic quantum number $m_l$ sets the orbital's orientation in space. For a given $l$ , it runs in integer steps from $-l$ to $+l$ — so there are $2l + 1$ orientations. An s subshell ( $l = 0$ ) has one orbital; p ( $l = 1$ ) has three ( $m_l = -1, 0, +1$ ); d ( $l = 2$ ) has five.

It is called 'magnetic' because these orientations have slightly different energies in a magnetic field — the Zeeman effect, which astronomers use to measure magnetic fields on the Sun and stars.

Electron shells labeled by n; the subshells (s, p, d) and their orientations come from l and m_l.

m_s: the spin quantum number

The spin quantum number $m_s$ describes the electron's intrinsic angular momentum — its spin — which can be $+1/2$ or $-1/2$ (in units of $\hbar$ ). Spin is not the electron literally spinning like a top; it is a built-in quantum property with no classical analogue.

Spin brings the most famous rule in chemistry, the Pauli exclusion principle: no two electrons in an atom may share the same four quantum numbers. Since $m_s$ has only two values, each orbital holds at most two electrons — with opposite spins. That single rule forces electrons into higher shells and builds the entire periodic table.

From quantum numbers to the periodic table

Electrons fill the lowest-energy states first (the Aufbau principle), two per orbital. Helium's two electrons share $(n=1, l=0, m_l=0)$ but differ in $m_s$ — one $+1/2$ , one $-1/2$ — satisfying Pauli. Carbon's six electrons read $1s^2 2s^2 2p^2$ : the 2p electrons spread across the three p orbitals with parallel spins (Hund's rule) before pairing up.

The periodic table's structure — two elements in row one, eight in row two, the s/p/d/f blocks — is simply this filling order written out. Quantum numbers are not abstract; they are the table on every classroom wall.

The validity checklist

For any electron: l < n, |m_l| ≤ l, and m_s = ±1/2. If a proposed set of quantum numbers breaks one of these rules, that state cannot exist.

Variables and units

Symbol	Name	Unit	Meaning
$n$	Principal quantum number	dimensionless	Energy level and rough size of the orbital (the shell)
$l$	Azimuthal quantum number	dimensionless	Orbital angular momentum; sets the orbital's shape (s, p, d, f)
$m_l$	Magnetic quantum number	dimensionless	Orientation of the orbital in space; −l … +l
$m_s$	Spin quantum number	dimensionless	Electron's intrinsic spin; only +1/2 or −1/2

Common mistakes

✗ Electrons orbit the nucleus like planets.

Orbitals are three-dimensional probability clouds. Quantum numbers describe the shape and energy of these clouds — not trajectories, which electrons do not have.

✗ Any combination of quantum numbers is allowed.

The rules are strict: l must be less than n, |m_l| cannot exceed l, and m_s is only ±1/2. A set like n = 1, l = 1 cannot exist.

✗ The Pauli exclusion principle applies only to electrons.

It applies to all fermions — every particle with half-integer spin, including protons, neutrons and quarks.

Real-world applications

Atomic spectroscopy identifies elements by the photons emitted when electrons drop between n levels — used in neon signs, lasers and stellar composition analysis.
The periodic table's rows and blocks are a direct map of the quantum-number filling order.
MRI machines manipulate proton spin states in magnetic fields to image soft tissue.

Frequently asked questions

What are the four quantum numbers?

n is the principal quantum number (energy level), l is the azimuthal quantum number (orbital shape), m_l is the magnetic quantum number (orbital orientation), and m_s is the spin quantum number (+1/2 or −1/2). Together they label a single electron's complete state in an atom.

What quantum numbers are possible for n = 2?

n = 2 allows l = 0 (the 2s subshell) and l = 1 (2p). For l = 0 there is only m_l = 0. For l = 1, m_l = −1, 0, +1 — three p orbitals. With two spin states each, the n = 2 shell holds 2 + 6 = 8 electrons, which is why the periodic table's second row has eight elements.

Why is the spin quantum number only ±1/2?

Spin is quantized like other angular momenta. Electrons are spin-1/2 fermions, so the component of spin measured along any axis can only be ±ħ/2 — written ±1/2 in units of ħ. Other particles differ: photons are spin-1, for example.

Do protons and neutrons have quantum numbers?

Yes — nucleons in a nucleus occupy shell-model states with their own quantum numbers, and the same Pauli principle applies. It explains the 'magic numbers' of extra-stable nuclei such as helium-4 and lead-208.